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                <h1 id="quantization-algorithms">Quantization Algorithms</h1>
<h2 id="symmetric-linear-quantization">Symmetric Linear Quantization</h2>
<p>In this method, a float value is quantized by multiplying with a numeric constant (the <strong>scale factor</strong>), hence it is <strong>Linear</strong>. We use a signed integer to represent the quantized range, with no quantization bias (or "offset") used. As a result, the floating-point range considered for quantization is <strong>symmetric</strong> with respect to zero.<br />
In the current implementation the scale factor is chosen so that the entire range of the floating-point tensor is quantized (we do not attempt to remove outliers).<br />
Let us denote the original floating-point tensor by <script type="math/tex">x_f</script>, the quantized tensor by <script type="math/tex">x_q</script>, the scale factor by <script type="math/tex">q_x</script> and the number of bits used for quantization by <script type="math/tex">n</script>. Then, we get:
<script type="math/tex; mode=display">q_x = \frac{2^{n-1}-1}{\max|x|}</script>
<script type="math/tex; mode=display">x_q = round(q_x x_f)</script>
(The <script type="math/tex">round</script> operation is round-to-nearest-integer)  </p>
<p>Let's see how a <strong>convolution</strong> or <strong>fully-connected (FC)</strong> layer is quantized using this method: (we denote input, output, weights and bias with  <script type="math/tex">x, y, w</script> and <script type="math/tex">b</script> respectively)
<script type="math/tex; mode=display">y_f = \sum{x_f w_f} + b_f = \sum{\frac{x_q}{q_x} \frac{w_q}{q_w}} + \frac{b_q}{q_b} = \frac{1}{q_x q_w} \sum{(x_q w_q + \frac{q_b}{q_x q_w}b_q)}</script>
<script type="math/tex; mode=display">y_q = round(q_y y_f) = round(\frac{q_y}{q_x q_w} \sum{(x_q w_q + \frac{q_b}{q_x q_w}b_q)})</script>
Note how the bias has to be re-scaled to match the scale of the summation.</p>
<h3 id="implementation">Implementation</h3>
<p>We've implemented <strong>convolution</strong> and <strong>FC</strong> using this method.  </p>
<ul>
<li>They are implemented by wrapping the existing PyTorch layers with quantization and de-quantization operations. That is - the computation is done on floating-point tensors, but the values themselves are restricted to integer values.  </li>
<li>All other layers are unaffected and are executed using their original FP32 implementation.  </li>
<li>For weights and bias the scale factor is determined once at quantization setup ("offline"), and for activations it is determined dynamically at runtime ("online").  </li>
<li><strong>Important note:</strong> Currently, this method is implemented as <strong>inference only</strong>, with no back-propagation functionality. Hence, it can only be used to quantize a pre-trained FP32 model, with no re-training. As such, using it with <script type="math/tex">n < 8</script> is likely to lead to severe accuracy degradation for any non-trivial workload.</li>
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